Quantitative observability for one-dimensional Schr\"odinger equations with potentials

Pei Su; Chenmin Sun; Xu Yuan

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arXiv · 2309.00963

Quantitative observability for one-dimensional Schr\"odinger equations with potentials

Pei Su·Chenmin Sun·Xu Yuan

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Abstract

In this note, we prove the quantitative observability with an explicit control cost for the 1D Schr\"odinger equation over $\mathbb{R}$ with real-valued, bounded continuous potential on thick sets. Our proof relies on different techniques for low-frequency and high-frequency estimates. In particular, we extend the large time observability result for the 1D free Schrodinger equation in Theorem 1.1 of Huang-Wang-Wang [20] to any short time. As another byproduct, we extend the spectral inequality of Lebeau-Moyano [27] for real-analytic potentials to bounded continuous potentials in the one-dimensional case.

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Pei Su, Chenmin Sun, Xu Yuan. 2023-09-02. Quantitative observability for one-dimensional Schr\"odinger equations with potentials. https://arxiv.org/abs/2309.00963

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